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Mirrors > Home > MPE Home > Th. List > oduleval | Structured version Visualization version GIF version |
Description: Value of the less-equal relation in an order dual structure. (Contributed by Stefan O'Rear, 29-Jan-2015.) |
Ref | Expression |
---|---|
oduval.d | ⊢ 𝐷 = (ODual‘𝑂) |
oduval.l | ⊢ ≤ = (le‘𝑂) |
Ref | Expression |
---|---|
oduleval | ⊢ ◡ ≤ = (le‘𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvex 6677 | . . . . 5 ⊢ (le‘𝑂) ∈ V | |
2 | 1 | cnvex 7624 | . . . 4 ⊢ ◡(le‘𝑂) ∈ V |
3 | pleid 16661 | . . . . 5 ⊢ le = Slot (le‘ndx) | |
4 | 3 | setsid 16532 | . . . 4 ⊢ ((𝑂 ∈ V ∧ ◡(le‘𝑂) ∈ V) → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
5 | 2, 4 | mpan2 689 | . . 3 ⊢ (𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
6 | 3 | str0 16529 | . . . 4 ⊢ ∅ = (le‘∅) |
7 | fvprc 6657 | . . . . . 6 ⊢ (¬ 𝑂 ∈ V → (le‘𝑂) = ∅) | |
8 | 7 | cnveqd 5740 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ◡∅) |
9 | cnv0 5993 | . . . . 5 ⊢ ◡∅ = ∅ | |
10 | 8, 9 | syl6eq 2872 | . . . 4 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = ∅) |
11 | reldmsets 16505 | . . . . . 6 ⊢ Rel dom sSet | |
12 | 11 | ovprc1 7189 | . . . . 5 ⊢ (¬ 𝑂 ∈ V → (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) = ∅) |
13 | 12 | fveq2d 6668 | . . . 4 ⊢ (¬ 𝑂 ∈ V → (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) = (le‘∅)) |
14 | 6, 10, 13 | 3eqtr4a 2882 | . . 3 ⊢ (¬ 𝑂 ∈ V → ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉))) |
15 | 5, 14 | pm2.61i 184 | . 2 ⊢ ◡(le‘𝑂) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
16 | oduval.l | . . 3 ⊢ ≤ = (le‘𝑂) | |
17 | 16 | cnveqi 5739 | . 2 ⊢ ◡ ≤ = ◡(le‘𝑂) |
18 | oduval.d | . . . 4 ⊢ 𝐷 = (ODual‘𝑂) | |
19 | eqid 2821 | . . . 4 ⊢ (le‘𝑂) = (le‘𝑂) | |
20 | 18, 19 | oduval 17734 | . . 3 ⊢ 𝐷 = (𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉) |
21 | 20 | fveq2i 6667 | . 2 ⊢ (le‘𝐷) = (le‘(𝑂 sSet 〈(le‘ndx), ◡(le‘𝑂)〉)) |
22 | 15, 17, 21 | 3eqtr4i 2854 | 1 ⊢ ◡ ≤ = (le‘𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ∅c0 4290 〈cop 4566 ◡ccnv 5548 ‘cfv 6349 (class class class)co 7150 ndxcnx 16474 sSet csts 16475 lecple 16566 ODualcodu 17732 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-ltxr 10674 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-dec 12093 df-ndx 16480 df-slot 16481 df-sets 16484 df-ple 16579 df-odu 17733 |
This theorem is referenced by: oduleg 17736 odupos 17739 oduposb 17740 oduglb 17743 odulub 17745 posglbd 17754 oduprs 30638 odutos 30645 ordtcnvNEW 31158 ordtrest2NEW 31161 |
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